The relevant question for the power of the king would be how much an additional king moving piece with no royal powers would be worth. If e.g. the Nb1 instead was a non royal king piece.
I did try something like this before, at least in weak engine play that piece was not very strong. Famously in the endgame the king is the "4 pawn piece", however, in the opening and midgame due to its low mobility it is less strong. For instance from b1 it would take moving the c pawn and then two king moves to even fight for the central squares; the knight can do it in one move.
My guess would be something like 2.5 pawns at the start of the game, and increasing in value as more and more pieces and pawns get traded.
I did try something like this before, at least in weak engine play that piece was not very strong. Famously in the endgame the king is the "4 pawn piece", however, in the opening and midgame due to its low mobility it is less strong. For instance from b1 it would take moving the c pawn and then two king moves to even fight for the central squares; the knight can do it in one move.
My guess would be something like 2.5 pawns at the start of the game, and increasing in value as more and more pieces and pawns get traded.